The Huovinen transform and rectifiability of measures

TL;DR

This paper proves that for sets of finite length, the existence of the Huovinen transform in principal value implies the set is rectifiable, linking a specific singular integral operator to geometric measure properties.

Contribution

It establishes a new rectifiability criterion based on the principal value existence of the Huovinen transform for sets of finite length.

Findings

Huovinen transform existence implies rectifiability

Connects singular integral operators with geometric measure theory

Provides a new criterion for rectifiability

Abstract

For a set $E$ of positive and finite length, we prove that if the Huovinen transform (the convolution operator with kernel $z^k/|z|^{k+1}$ for an odd number $k$) associated to $E$ exists in principal value, then $E$ is rectifiable.